The Beast That Ascendeth
We know that the beast is formed from two separate demonic factions, pictured in the fifth and sixth trumpets. Satan is attempting to construct the symmetric group S5 upon the five refining churches, allowing a "wider path" (more elements and therefore permutations) and a "closed" system as opposed to the intent of the creator shown in the cyclic group C7.
The five elements of the G-Set upon which S5 acts are composed from the four earthly elements (the coset of the static subgroup) as well as the fifth element, (which is actually not one of the "churches") a fallen star with the key to the bottomless pit. The beast that ascendeth requires a field of characteristic two, and an additive subgroup of that field with an orbit under multiplication of length two.
Now, this is impossible for any finite field, since we would require a multiplicative subgroup of index 2; but this would make the frobenius map a proper morphism on the multiplicative group of the field. What we must therefore have is an algebraic closure where we can construct just such an additive group. What then?
We must attach a root of multiplicative order two; yet we can drop the requirement that the additive group need be of index 2, if the algebraic closure is infinite! We can continue in somewhat the same matter as would Hamilton, whom postulated the existence of the quaternions. Likewise this is an old trick; we extend the field by the "negative square root of one" with an ordered cartesian pair.
Therefore (a,b) becomes as it were "a - b".
Then we can keep our algebraic closure if we can adjoin a separate system based on an order two element "f", which will form our "-" so that a+ bf acts as if it were a - b. Clearly we need to define our addition on (a,b) and we may do this in this manner: (a,c)+(b,d)=(a+b,c+d) and so multiplication acts in the following manner (a,c)*(b,d) = ((a*b+c*d),(a*d+b*c)). Note that the subfield formed of (x,0) for all x from our algebraic closure is isomorphic to the algebraic closure itself.
All we need now is to identify our coordinate pair: We form these from the serpent spirits of the sixth trumpet, where the "negative element" "f" is a simple transposition of two elements in the octal. (we only need a transposition and it can be of any numbers of pairs.)
We see the construction using the serpent spirits of the sixth trump is identical in principle to the one found here. The amalgam of these two systems allows the construction of the full group S5 upon the four earthly elements with the inclusion of the fallen star. This star obviously had the key to solving the problem as above.
Now we must ask if we require our field to be infinite...Do we need an algebraic closure after all? We certainly do have a closed field from which to form these "orbits" We do need to show that we have an orbit of length two on every element. Clearly we may choose an additive subgroup of index two,.. but it becomes clear that the morphism conditions will not hold.
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